   Chapter 8.3, Problem 30E

Chapter
Section
Textbook Problem

# Finding an Indefinite Integral Involving Secant and Tangent In Exercises 21–34, find the indefinite integral. ∫ tan 3 3 x d x

To determine

To calculate: The value of the following indefinite integral tan33xdx

Explanation

Given: The provided indefinite integral is tan33xdx

Formula Used:

The following trigonometric identity,

sec23x1=tan23x

The following differential formula,

ddxtanx=sec2x

The following integral formulas,

yndy=yn+1n+1+C.

dzz=ln|z|+C

Calculation:

Consider the provided integral, tan33xdx.

tan33xdx=tan23xtan3xdxtan33xdx=(sec23x1)tan3xdx.

tan33xdx=sec23xtan3xdxtan3xdx ------(1)

Now integrate this equation by the method of substitution,

Let consider, tan3x=y ------(2)

Now differentiate equation (2) with respect to x, we get

ddxtan3x=dydx.

sec23xddx(3x)=dydx.

3sec23xdx=dy.

sec23xdx=13dy ------(3)

Now substitute the values from equation (2) and (3) to equation (1), we get

tan33xdx=13(y)dysin3xcos3xdx

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